MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
115 
variation such as we have just given. No doubt, if we were to do this, we should at 
last get a curve differing from the original one, and, since Y nn is the same if x be the 
same, it seems that we should get the variations all of the same sign, and that the 
result could thus at once be extended to a finite length of the curve; but, although it 
would be true that the sign of the second variation in passing from any one of the 
curves thus found to the consecutive one would have the same sign as Y nn dx, yet the 
first variation would not vanish. 
It is not difficult to see that these two alternative conditions, treated of in Cases 
1 and 2 respectively, are independent. For, if the points obtained by the two criteria 
coincided, the least root of the determinant equation (ct) in the note # must coincide 
with the least value of x for which Y, m = 0, and it seems evident that there can be 
no such connection. It may, however, be as well to give an example, to show that 
the condition in Case 2 may cease to be fulfilled, while that in Case 1 still holds. 
Taking for U the expression j ifix dx, we have 
§ U = 4 j Sy y ?, x dx = [4 y z x Sy] 1 — 4 [ — (y 3 x) Sy dx, 
0 J CCOC 
and the equation given by the calculus is 
d •„ 
— y A x — 0 : 
dx J 
integrating, we get 
y — cxY s -j- c', 
whence it is easy to see that Y nn dx or 4.3 .y 2 x dx changes sign as x passes through 
the value zero. (It might be thought sufficient to say that, as y 3 cannot change sign, 
* If y = f (x, Cj, c 2 , ■ ■ . c 2n ) be the general solution obtained by making 
T »-§ + e= 0 ' 
then, denoting d//dc 15 df/dc 2 , &c., by y v y 2 , &c., it is well known that we have for the value of aq, to 
which a second curve of the species can be drawn from x 0 , so as to have, at x 0 and the point we seek, 
contact of the (n — 2) tl ‘ order with the curve y = / (x, c v c 2 , . . . c 2n ), the equation 
y' i» 
y'v • 
. . . y' 29! 
Vv 
2/3. 
. . . . y 2n 
• 9 
• 9 * 
. 
(»—1) 
y i > 
(m—1) 
y 3 . 
,/ (n-l) 
. . y hi 
2/19 
2/3. 
.... y m 
2/i. 
2/3. • 
y 2 n 
• » 
* > 
7/ (W—1 
V\ ? 
2/2 (K-1) 9 
.... yJ n - 2) 
where y' means the value of y when x 0 is substituted for x. The least value of x satisfying this, or more 
properly the value first reached in going from x 0 via, the curve, gives the point in question. 
Q 2 
